A Conjecture on Fields of Extremals with Slopes Diverging (Open)

Summary. In the study of the extremals of a functional of the type
$F(z)=\int_{0}^{T}L(t,z(t),z^{\prime }(t))dt$, the following situation
often arises: the extremals $f_{\lambda }$ that satisfy $f_{\lambda }
(0)=a$ and $f_{\lambda }^{\prime }(0)=\lambda$ constitute a central field,
and the slope of the field at each point diverges, i.e., $\lim_{\lambda
\rightarrow \pm \infty } f_{\lambda}^{\prime }(t)=\pm \infty $. Under
these conditions, we conjecture that $\lim\limits_{\lambda \rightarrow
\pm \infty} f_{\lambda}(T)=\pm \infty $ and, hence, that an extremal will
exist for any condition at the end point.